Math 237B (Topics in Differential Equations)

Course Topics: Mathematical and Computational Geometric PDE
Instructor: Prof. Michael Holst (5739 AP&M, mholst@math.ucsd.edu)
Term: Spring 2007
Lecture: 9:30a-10:50a Tu-Th, APM 5402 (MOVED FROM 5829)
Textbook(s):

N. Straumann, General Relativity: With Applications to Astrophysics, Springer, 2004.
R. Wald, General Relativity, Univ. of Chicago Press, 1984.
A. Cohen, Numerical Analysis of Wavelet Methods, JAI Press, 2003.
Class webpage: http://ccom.ucsd.edu/~mholst/teaching/ucsd/237b_s07

CATALOG DESCRIPTION: 237A-B-C. Topics in Differential Equations (4-4-4)
May be repeated for credit with consent of adviser. Prerequisite: consent of instructor.
GRADES, EXAMS, DATES: Your grade in the course is based on attending the lectures and participating in the class.
  • Final: Thursday 13 June 2007 (9am-11am, 5402 APM)
  • No Class: April 10, April 12, May 15
  • First lecture: Tuesday 3 April 2007.
  • Last lecture: Thursday 7 June 2007.
  • Finals week: 11-15 June 2007.
ANNOUNCEMENTS, NOTES, ETC:
  • IPAM Workshop Notes on Elliptic PDE and Approximation theory
  • Notes on Calculus in Banach Spaces, Variational Methods, and Mechanics
SCHEDULE FOR THE LECTURE TOPICS: The overall plan is to first develop the basic mathematical framework for understanding a class of PDE arising in geometric settings, and based on this develop a general approximation theory framework for designing provably good computational methods for this class of PDE.

Relevant applications include models of biomembranes, models of fluid interactions, geometric flows in material science, flows of metrics in geometry, models of black hole collisions in astrophysics, and physics-based reality models in computer animation.

Since we only have ten weeks, we will build the lectures primarily around one particular geometric PDE model (general relativity) describing the gravitational interaction of massive objects such as black holes and neutron stars. However, the notation, theory, and techniques we develop will be useful for other applications.

The outline below will be updated as the quarter proceeds, based on how the interests of the participants and the lecturer evolve. We will leverage much of the material developed last quarter in 237A, but the course will again be basically self-contained.

Lecture Topics Covered
1 (4/3/07, week 1) Differentiable manifolds, connections, covariant differentiation, tangent vectors, differential forms, tangent space, cotangent space, Hausdorff property and partition of unity.
2 (4/5/07, week 1) Fiber bundles, tangent and cotangent bundles, tangent and cotangent spaces as fibers of a bundle, tensor fields as sections of a bundle, jet bundles.
3 (4/17/07, week 3) Lie derivative, exterior derivative, integration, manifolds with boundary, Stokes theorem, divergence theorem.
4 (4/19/07, week 3) Metrics, Riemannian and pseudo-Riemanian manifolds, Levi-Cevita connection, Christoffel symbols.
5 (4/24/07, week 4) Laplace-Beltrami operator, properties of the Laplace-Beltrami operator, weak maximum principle, a priori L-infinity bounds, sub- and super-solutions.
6 (4/26/07, week 4) General nonlinear geometric elliptic operators and PDE, properties of a semilinear Laplace-Beltrami-like operator, weak maximum principle, a priori L-infinity bounds, sub- and super-solutions.
7 (5/1/07, week 5) Lecture 1 on G. Dziuk paper (Lectures Notes in Math 1357, 1988) on a priori error estimates for the Laplace-Beltrami operator equation.
8 (5/3/07, week 5) Lecture 2 on G. Dziuk paper (Lectures Notes in Math 1357, 1988) on a priori error estimates for the Laplace-Beltrami operator equation.
9 (5/8/07, week 6) Lecture 3 on G. Dziuk paper (Lectures Notes in Math 1357, 1988) on a priori error estimates for the Laplace-Beltrami operator equation.
10 (5/10/07, week 6) Lecture 1 on M. Holst paper (AiCM, 2001) on a priori and a posteriori estimates for a general class of semilinear geometric PDE, with application to the constraints in GR.
11 (5/15/07, week 7) Lecture 2 on M. Holst paper (AiCM, 2001) on a priori and a posteriori estimates for a general class of semilinear geometric PDE, with application to the constraints in GR.
12 (5/17/07, week 7) Lecture 3 on M. Holst paper (AiCM, 2001) on a priori and a posteriori estimates for a general class of semilinear geometric PDE, with application to the constraints in GR.
13 (5/22/07, week 8) Lecture 1 on A. Demlow and G. Dziuk paper (SINUM, 2006) on a posteriori estimates for the Laplace-Beltrami operator equation.
14 (5/24/07, week 8) Lecture 2 on A. Demlow and G. Dziuk paper (SINUM, 2006) on a posteriori estimates for the Laplace-Beltrami operator equation.
15 (5/29/07, week 9) Lecture 3 on A. Demlow and G. Dziuk paper (SINUM, 2006) on a posteriori estimates for the Laplace-Beltrami operator equation.
16 (5/31/07, week 9) Lecture 1 on G. Dziuk and C. Elliot paper (IMA J. Numer. Anal., 2006) on a priori estimates for finite elements on evolving surfaces.
17 (6/5/07, week 10) Lecture 2 on G. Dziuk and C. Elliot paper (IMA J. Numer. Anal., 2006) on a priori estimates for finite elements on evolving surfaces.
18 (6/7/07, week 10) Lecture 3 on G. Dziuk and C. Elliot paper (IMA J. Numer. Anal., 2006) on a priori estimates for finite elements on evolving surfaces.
19 (6/13/07, Final Lecture) Overview of Huisken-Ilmanen paper on inverse mean curvature flow, Hamilton paper on Ricci flow, and prospects for explicit (rather than implicit) and intrinsic (rather than extrinsic) numerical treatment.